for each of these solutions we need to nd a

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Unformatted text preview: ossible to find explicit solutions to these partial differential equations under the simplest boundary conditions. For example, the general solution to the one-dimensional wave equation ∂2u ∂2u = c2 2 ∂t2 ∂x for the vibrations of an infinitely long string, is u(x, t) = f (x + ct) + g (x − ct), where f and g are arbitrary well-behaved functions of a single variable. Slightly more complicated cases require the technique of “separation of variables” together with Fourier analysis, as we studied before. Separation of variables reduces these partial differential equations to linear ordinary differential equations, often with variable coefficients. For example, to find the explicit solution to the heat equation in a circular room, we will see that it is necessary to solve Bessel’s equation. The most complicated cases cannot be solved by separation of variables, and one must resort to numerical methods, together with sufficient theory to understand the qualitative behaviour of the s...
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This document was uploaded on 01/12/2014.

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